Persistence in the shadow of killers.

mm m

MICROBIOLOGY

ORIGINAL RESEARCH ARTICLE

published: 14 July 2014
doi: 10.3389/fmicb. 2014. 00342

Persistence in the shadow of killers

Robert M. Sinclair*

Mathematical Biology Unit, Okinawa Institute of Science and Technology, Okinawa, Japan

Edited by:

Luis Raul Comolli, Lawrence
Berkeley National Laboratory, USA

Reviewed by:

Chiara Mocenni, University of Siena,
Italy

Franz Luef, NTNU Trondlheim,
Norway

'Correspondence:

Robert M. Sinclair, Mathematical
Biology Unit, Okinawa Institute of
Science and Technology Graduate
University, 1919-1 Tancha,
Onna-son, Okinawa 904-0495,
Japan

e-mail: sinclair@oist.jp

Killing is perhaps the most definite form of communication possible. Microbes such as
yeasts and gut bacteria have been shown to exhibit killer phenotypes. The killer strains are
able to kill other microbes occupying the same ecological niche, and do so with impunity.
It would therefore be expected that, wherever a killer phenotype has arisen, all members
of the population would soon be killers or dead. Surprisingly, (1 ) one can find both killer and
sensitive strains in coexistence, both in the wild and in in vitro experiments, and (2) the
absolute fitness cost of the killer phenotype often seems to be very small. We present an
explicit model of such coexistence in a fragmented or discrete environment. A killer strain
may kill all sensitive cells in one patch (one piece of rotting fruit, one cave or one human
gut, for example), allowing sensitives to exist only in the absence of killer strains on the
same patch. In our model, populations spread easily between patches, but in a stochastic
manner: one can imagine spores borne by the wind over a field of untended apple trees, or
enteric disease transmission in a region in which travel is effectively unrestricted. What we
show is that coexistence is not only possible, but that it is possible even if the absolute
fitness advantage of the sensitive strain over the killer strain is arbitrarily small. We do
this by performing a specifically targeted mathematical analysis on our model, rather than
via simulations. Our model does not assume large population densities, and may thus be
useful in the context of understanding the ecology of extreme environments.

Keywords: intra-species interaction, killer phenotype, competitive exclusion, coexistence, mathematical analysis

INTRODUCTION

The Purpose of Computing Is Insight, Not Numbers (Hamming,
1987).

It will be useful to begin with a clear statement of what this work
is really about. The central question being addressed is whether
killer and sensitive phenotypes could in theory coexist in the same
environment even if the absolute fitness penalty for the killer
phenotype were arbitrarily small. The question is motivated by
experimental and field observations to be described below, but
is to be approached here from a theoretical point of view. This
immediately implies that what is needed is a theoretical frame-
work which has two apparently contradictory properties: First, it
must capture important aspects of the relevant biology. Second,
it must be simple enough that the question can be answered.
This theoretical framework will therefore necessarily be a com-
promise between realism and tractability. Furthermore, what is
actually needed is only a single affirmative result, since that proves
the theoretical possibility of coexistence. The phrase "arbitrar-
ily small" excludes some standard approaches, and it is best to
make this point now rather than later, because it is this com-
pact phrase which drags us into what will be unfamiliar territory
for many readers. If the theoretical framework were to be in the
form of a standard numerical simulation of a model, then the
best one could do would be to show that coexistence is possible
for given small absolute fitness penalties for the killer pheno-
type. This is not the same as "arbitrarily small": if a simulation

were to show coexistence to be possible for a fitness penalty of
0.1, it would still leave open the question of whether coexistence
could be possible for a fitness penalty of 0.01 and so on ad infini-
tum. We are thus motivated, by the question we have chosen to
investigate, to search outside of the box of standard scientific com-
puting tools until a truly suitable approach is found. The field of
mathematical analysis (Ross, 1980) offers itself at this point, since
it includes powerful techniques for dealing with the arbitrarily
small. Moreover, we can apply elementary methods of mathe-
matical analysis to mathematical models. We are now ready to
sketch our approach: we will analyse a model which captures
much of the biology but is simple enough to allow a mathemati-
cal analysis to be performed. If we can show that coexistence of
killer and sensitive phenotypes is possible, for arbitrarily small
absolute fitness penalties for the killer phenotype, for this single
model, then we will have answered our question in the affirma-
tive. Once this has been done, the particular model is no longer
of direct importance (in the same sense that a certain telescope
may be used to make an important astronomical observation,
but it is almost always the observation which has lasting impor-
tance, not the telescope), although we hope it may be useful in
other investigations. To paraphrase Hamming, the purpose of
our analysis is insight, not the establishment of a computational
model per se. In other words, the reader should not be expecting
to see the standard modeling approach of computational biol-
ogy, with all of the standard parameter fitting and graph plotting
that entails. Instead, the reader should expect to find here some-
thing rather unusual, a more truly mathematical approach, but

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July 2014 | Volume 5 [ Article 342 | 1

Sinclair

In the shadow of killers

something which has already been proposed in a related context
(Silva, 2011).

The phenomenon of killer phenotypes, which possess the abil-
ity to kill conspecifics while being themselves immune (Marquina
et al., 2002; Breinig et al., 2006), is widespread in the microbial
world (Schmitt and Breinig, 2006; Schrallhammer, 2010; Holt
et al., 2013). As more such systems are studied, it is becoming
increasingly clear that the evolutionary contexts are so varied
(Cornejo et al., 2009 provide a surprising example) that it may
be impossible to encompass all that is relevant (such as biodiver-
sity Czaran et al., 2002) in a single, simple model. Classical theory
predicts that competition for a single resource should result in
the survival of only one competitor (Hardin, 1960), and yet sensi-
tive strains can be more common than killers (Riley and Gordon,
1999; Pieczynska et al., 2013), and it has been observed that there
can be coexistence between killer and sensitive strains, necessi-
tating the development of new models (Czaran and Hoekstra,
2003; Vadasz et al., 2003). Here, we provide a novel explicitly
solvable mathematical model of the population dynamics of a
species with killer and sensitive strains inhabiting a fragmented
but potentially highly interconnected environment. Our model
includes only killer and sensitive phenotypes. While we were orig-
inally inspired by the image of fallen, over-ripe fruits beneath a
grove of fruit trees, with spores providing the mechanism of con-
nectivity, the mathematical structure of the model allows it to be
applied to many other situations: enteric pathogens live in iso-
lated environments (within the digestive tracts of their individual
hosts), but transmission between hosts does occur and can repre-
sent a high degree of connectivity in the case of a pandemic. Also,
intraterrestrial microbial communities living in largely isolated
caves or niches may be sporadically connected by flooding events
(Hawes, 1939), as could psychrophilic microbial communities
living in niches in or on ice (Margesin and Miteva, 2011) be con-
nected during annual thawing or via other dispersal mechanisms.

Rather surprisingly, it has been shown that the cost of toxin
production can be negligible (Garbeva et al., 2011), and is pre-
sumably only a few percent when measurable (Wloch-Salamon
et al., 2008). We asked whether, in a model, coexistence of killer
and sensitive phenotypes is possible for any difference in absolute
fitness between killer and sensitive phenotypes, however small.
That requires analysis rather than simulation, and this point has
therefore decisively influenced our approach.

MATERIALS AND METHODS

We describe here an explicitly solvable model of yeast population
dynamics on an infinite number of patches, in which killer and
sensitive strains can coexist. Our model includes killer and sen-
sitive strains only. In the following, we will use the example of a
killer yeast in our verbal descriptions of the model. A full mathe-
matical treatment would not be appropriate here. We will instead
provide what may be called a sketch of the model and our analy-
sis of it. The Supplementary Material contains details of the most
important part of the mathematical analysis, but it is also best
described as a sketch rather than a formal proof.

Each patch is intended to represent a single piece of fruit. A
patch can be colonized by spores from any patch. If a patch is col-
onized only by spores of the sensitive yeast strain, then the patch

will emit only spores of the sensitive strain. If a killer yeast spore
lands on a patch, then any sensitive yeast colony will be eradi-
cated, and the patch will emit only spores of the killer yeast. If a
sensitive yeast spore lands on a patch colonized by killer yeast, it
will not survive nor influence the (killer yeast) spore production
of the patch. The number of spores emitted by a patch depends
only upon the type of yeast that has successfully colonized it. If
no spores have landed on a patch then that patch will emit no
spores. Sporulation occurs in all patches simultaneously, leaving
all patches barren and ready for the next cycle, initialized by the
dispersal of the spores.

Let fs > 1 and fa > 1 denote the average number of spores
emitted per patch colonized by sensitive (S) or killer (K) strains,
where these are to be understood as effective rather than abso-
lute values, since the model assumes that all spores are viable and
eventually find a patch. As expected, these numbers play the role
of absolute fitnesses. Also, let 0 < xs < 1 and 0 < xk < 1 denote
the respective fractions of patches successfully colonized (at the
time of sporulation) by the two strains.

The dynamics of the killer yeast strain is not in any way influ-
enced or restricted by the sensitive strain, and so can be treated
independently. The probability of a given patch not being reached
by any killer strain spore is

The reason for this can be understood by first considering a finite
number of patches, and then taking the limit as that number
goes to infinity. Let n denote the (finite) number of patches. The
probability that any given spore will not land on any given patch
is 1 — 1/f), assuming random dispersal. The total number of col-
onized patches is n xk, so the average number of spores emitted
in total is h/kXk- The probability that none of these land on any
given patch is (1 - l/n) nfKXK . If we now let n go to infinity, we
find that we can quite directly use one of the standard definitions
of the exponential function:

lim 1

ii/kxk

lim

\-\Ikxk

-fK*K

This will be, for an infinite number of patches, the fraction of
patches which are not reached by any spore. On the other hand,
the fraction of patches which are reached by a spore must be the
remainder, or 1 — e~f KXK . The killer strain population dynamics
is therefore described by the map

x K I

(i _ e -kxA

(1)

Since we can write

xk h» f K x K + O (f|%|) ,

we can state that fa plays the role of absolute fitness for small
/kXk • In other words, our model includes the phenomenon of

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In the shadow of killers

exponential growth when resources are not a limiting factor, and
this exponential growth can be used to define an absolute fitness.

According to standard theory, the map (Equation 1) has an
unstable fixed point at xk = 0 and a stable fixed point at

x K =X K = l +

W(-f K e-k)
k

where W is the principal branch of the Lambert W function
(Corless et al., 1996).

The dynamics of the sensitive strain is governed by the same
equations in the complete absence of spores of the killer strain.
The reason for this assumption is the observation (discussed
above) that the difference in absolute fitness between the killer
and sensitive strains can be very small. In the presence of an
established killer strain population occupying a fixed fraction
{Xk ) of all patches, the sensitive strain population dynamics is
determined by the map

X S !

(l - e^ xs ) (1

X K ).

which has an unstable fixed point at xs = 0 and a stable fixed
point at

x s =X s =(l-X K ) +

W(-(l-X K )f s e- {l - x ^)
fs

if and only if

fs>

-k

W(-fce-f*y

(2)

One can construct (details are in the Supplementary Material and
see also Figure 1) the upper bounds

5fr

4 >

f-
Jk

2k

-k

W(-f K e-k)

(3)

for 1 < fa < 1.09. If the cost of the killer phenotype is S > 0, so
that fx = fs — S, and we set fs = 5k — 4, then for any very small
choice of 0 < S < 0.36, we can state that coexistence is possible in
our model, and can provide an explicit family of examples, with
f K = 1 + 8/4 and/s = 1 + 55/4.

Given a stable subpopulation of the killer strain, a necessary
condition for establishment of a subpopulation of the sensitive
strain from a finite number of sensitive spores is

fs(l-X K ) > 1,

which is identical to the previous inequality (Equation 2). We
omit the technical details here, but the product fs (1 — Xk) is the
effective absolute fitness of the sensitive strain in the presence of
an established population of the killer strain. For each patch suc-
cessfully colonized by a sensitive strain, an average of/5 spores will
be emitted, but only 1 — Xk of patches are free of the killer strain,
so only this fraction is will survive to sporulation. These consid-
erations apply equally to an initial exponential growth phase or a

1 2 3 4 5 6 7 8

K

FIGURE 1 I A comparison of the composition of an exponential
function and the Lambert W function, W ( — f K e~ ,K ), and the upper
bound derived in the Supplementary Material, both plotted as
functions of fx > 1. Note that both curves are monotonically increasing, a
property which facilitates analysis.

stable state, and therefore the agreement with Equation (2) is to
be expected.

The total fraction of patches stably colonized by either strain is

XK+Xs=l+ n-s^if^^ <h

Note that

fs

Xk + Xs > Xk

if Equation (2) is satisfied, meaning that the sensitive strain, when
present, only contributes to total population.

Can any ratio of killer to sensitive phenotypes be achieved in
this model? Furthermore, can any total fraction of patches be sta-
bly colonized? Since Xk and Xs are continuous functions of fie
and/s, an d

lim X K = 1,

fK OO

lim X s = 1 - X K ,

lim Xk = lim X s = 0

fc^O f S ^-f K /W(-f K c-fK)

and

lim (X K + X s ) = lim (X K + X s ) = 1,

all pairs (Xk , Xs) for which Xk + Xs < 1 holds can be achieved
by suitable choices offc or f$.

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July 2014 | Volume 5 | Article 342 | 3

Sinclair

In the shadow of killers

As a numerical example, if = 2 and fs = 5, then Equation
(2) is satisfied, and two subpopulations of sizes Xk % 0.797 (i.e.,
79.7% of patches) and X s ~ 0.006 (i.e., 0.6% of patches) can sta-
bly coexist. If one were interested in trying to fit a minimalist
model of this type to real data, note that it would not be enough to
know the ratio of killer-dominated patches to sensitive-colonized
patches. One also needs the fraction of patches which are colo-
nized by neither strain, data which is not always reported in the
literature.

RESULTS

Two direct consequences of the model are (1) that killer and sen-
sitive strains can coexist in any given proportion, and (2) that the
presence of a sensitive subpopulation increases the total popula-
tion size of yeast (including both strains) without reducing the
population size of the virus population maintained by the killer
yeast strain. Taking a broader point of view, the second conse-
quence means that the species benefits from having both sensitive
and killer strains.

COEXISTENCE FOR ARBITRARILY SMALL COST OF KILLER PHENOTYPE

Since our model is explicitly solvable, we are able to perform
a mathematical analysis which showed (see the Equation 3
and the associated comments above) that coexistence is possi-
ble for any extra fitness cost of the killer phenotype, however
small.

As a numerical example, if fs = 1.0001, then we have coex-
istence for/jf = 1.000049, and the very small fitness cost repre-
sented by S = 0.000051. The corresponding fractions are Xk ^
0.0000996 and X s ~ 0.00000399. One notices that very small dif-
ferences in fitness are achieved by populations for which the total
fraction of colonized patches is also very small. This is the rea-
son to suggest that this model may best be suited to extreme
environments.

Using the explicit formulae from our analysis, /k = 1 + 5/4
and f s = 1 + 55/4, for a target cost of 8 = 0.00004, we have
f K = 1.00001 and fs = 1.00005, with the respective fractions
being X K » 0.00002 and X s ~ 0.00006. Here we see the power
of the analysis: we are able to construct infinitely many fur-
ther such examples for even smaller values of i5, without lower
limit.

Therefore, we are able to construct pairs (/k,/s) for which
coexistence is guaranteed, and, furthermore, do so for any given
fitness cost S for the killer phenotype, however small.

DISCUSSION

It is not intuitively obvious that sensitive strains can sur-
vive in the presence of killers, given that our model has no
fixed barriers to prevent the sensitive strains from being erad-
icated by encounters with killers. The value of our model
lies not only in this prediction, which is consistent with
other, related, models (the semi-analytical configuration-field
approximations for the one- and two-species SCA models of
Czaran and Hoekstra, 2003 in particular), but also in the
fact that it is explicitly solvable, a property which allows
types of analysis to be performed which are truly comple-
mentary to what is possible with simulations alone (Silva,
2011).

We have been able to prove that killer-sensitive coexistence is
possible for any fitness penalty of the killer phenotype, however
small. This is important because it has been shown that there
does not have to be any measurable fitness cost for antibiotic pro-
duction (Garbeva et al., 2011). The fact that our model applies
naturally to communities with low total population densities
suggests that it may be applicable to microbial communities in
extreme environments.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online
at: http://www.frontiersin.org/journal/10.3389/fmicb.2014.
00342/abstract

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Conflict of Interest Statement: The author declares that the research was con-
ducted in the absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.

Received: 31 March 2014; accepted: 20 June 2014; published online: 14 July 2014.
Citation: Sinclair RM (2014) Persistence in the shadow of killers. Front. Microbiol
5:342. doi: 10.3389/fmicb.2014.00342

This article was submitted to Terrestrial Microbiology, a section of the journal Frontiers
in Microbiology.

Copyright © 2014 Sinclair. This is an open-access article distributed under the terms
of the Creative Commons Attribution License (CC BY). The use, distribution or repro-
duction in other forums is permitted, provided the original author(s) or licensor are
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